Blipped trajectories have been, tried, with limited success, in MRI. Magnetic resonance imaging (MRI) sequences are characterized by both radio frequency (RF) pulses and time-varying gradient magnetic fields. RF pulses are used to align resonant nuclei to facilitate generating a measurable signal. Gradient fields are used to spatially encode signals so that signals coming from one excited location can be distinguished from signals coming from another location. The signals are collected and mapped into an array. The array may be referred to as a “k-space” array. K-space represents the spatial frequency content of the imaged object. MRI gradient fields determine the location in the k-space array for a data point. The order in which k-space points are acquired is determined by a k-space trajectory. One type of k-space trajectory is a blipped trajectory. Some blipped trajectories are illustrated in FIG. 1.
MRI involves controlling gradients to produce specific magnetic fields at specific locations at specific times. While gradients can be turned on and off, there are limits on how quickly the gradients can be turned on and off. Therefore much attention has been paid to how and when to manipulate gradients to produce useful, efficient trajectories through k-space. The efficiency becomes more important as shorter and shorter acquisition times are sought in MRI.
In a basic example, consider a two dimensional rectangular slice as illustrated in FIG. 2. The rectangular slice 100 can be divided logically into a set of regions that may be referred to as pixels or voxels. FIG. 2 illustrates slice 100 divided into pixels P1 through P25. Slice 100 can be seen to lie in an X-Y plane as indicated.
One conventional rectilinear trajectory creates conditions in the pixels P1 through P25 in order by changing the X gradient to move from P1 to P2 to P3 to P4 to P5 and then changing the Y gradient to move from P5 to P6 and then changing the X gradient to move from P6 to P7 to P8 to P9 to P10 and then changing the Y gradient to move from P10 to P11, and so on until all twenty five pixels have been traversed. This simple rectilinear trajectory is intuitively obvious and attractive. However, in some cases, this simple rectilinear trajectory may produce sub-optimal results in MRI. The sub-optimal results may be associated with, for example, how long it takes to sample the entire rectangle 100, how frequently the center of k-space is visited, interference from neighboring pixels, and other factors. The sub-optimal results may also be associated with, for example, the regularity in k-space between pixel acquisitions when signal is acquired sequentially from one pixel and then from a neighboring pixel using substantially similar conditions.
Therefore, non-rectilinear trajectories (e.g., radial, spiral) have been developed. These trajectories have been employed to support recent acquisition strategies (e.g., compressed sensing) associated with acquiring signals from moving objects. Imagine photographing a moving object (e.g., spiked volleyball). If you have a camera with a slow shutter speed and slow film, the spiked volleyball will be a blur. But if you have a camera with a faster shutter speed, the volleyball may be more clear. If you have a camera with a fast enough shutter and fast enough film, you may even be able to freeze the spiked volleyball in mid-flight. In photography, the image clarity of a moving object is directly related to “shutter speed”.
In MRI, to improve the imaging of moving objects (e.g., heart, blood), it may also be desirable to have a faster “shutter speed”, which is achieved by reducing the amount of time that it takes to acquire signal from the moving object. One way to reduce the amount of time that it takes to acquire signal is to perform rapid incoherent sampling using, for example, a compressed sensing approach.
Signal processing has generally accepted the assumption that a signal should be sampled at a rate of at least twice its highest frequency in order to be represented without error. However, this assumption may not be valid in some cases. Additionally, the assumption may lead to unnecessarily high sampling rates in some cases. Consider that much signal processing involves compressing data soon or immediately after sensing. The compression balances signal representation complexity against error. If a signal is going to be compressed immediately after sensing anyway, then it may not make sense to perform the full sensing. Instead, a compressed sensing may make sense.
Compressed sensing may involve sampling a signal at a reduced rate and performing compression soon after sampling. In MRI, the number of measurements taken is proportional to the scan time. Reducing the number of points sampled, as occurs in compressed sampling, reduces the scan time. However, reducing the number of points sampled leaves “holes” in the sampled data. The holes may be filled by extrapolating missing sample points from acquired sample points to produce a full data set from which an image can be made. Missing samples can be extrapolated by enforcing sparseness in a transform. However, the under-sampling may produce under-sampling artifacts.
Compressed sensing may rely on redundancy in signals. One useful redundancy exists when a signal is sparse, meaning that the signal has many coefficients close to or equal to zero when represented in some domain. Incoherent sampling seeks to have under-sampling artifacts be incoherent within the object so that they appear more like noise and less like signal. While there has been some success in incoherent sampling in three dimensions (3D), there has been less success in incoherent sampling in two dimensions (2D), particularly in compressed sensing approaches. Since MRI apparatus are frequently configured to acquire 2D “slices”, some MRI apparatus may have sub-optimal results when attempting incoherent sampling with compressive sensing.